To define ‘b is an ancestor of c’:
~(b=c) & (x){ c mem x & ((y)(z)((z mem x & Parent yz) -> y mem x) -> b mem x}
(Goodman and Quine "Steps Towards a Constructive Nominalism" JSL, Vol. 12, No. 4, 1947. p. 108)
I am going to simplify the discussion, somewhat, by at first using a predicative approach rather than a class theoretic approach. I will do this only in order to establish the transitivity of the ancestral. I will then proceed to
examine an option which dispenses with the class notation. My point here is going to be that whereas there
at first appears to be a draw back to treating the ancestral in terms of classes there is an advantage in doing
so, particularly, in applying the concept to the issue of causation. The main objection to the class theoretic
approach is that in the above formulation our only constraint on the classes is the class, x, contains the parents
of all members of x. So anything else can be included as long as this is not disturbed. With respect to ‘x’ as
Quine puts it, much else may be included: "ancestors, and neckties; for, neckties being parentless, their inclusion
does not disturb the fact that that all parents of members are member." (Methods of Logic, 1972, p. 238)
Fc & (Fx & (Hyx --> Fy) --> Fb (cRb)
Fb & (Fx & (Hyx --> Fy) --> Fd (bRd)
Therefore,
Fc & (Fx & (Hyx --> Fy) --> Fd (cRd)
Proof:
1. Fc & (Fx & (Hyx --> Fy) --> Fb Assumption
2. Fb & (Fx & (Hyx --> Fy) --> Fd Assumption
3. Fc & (Fx & (Hyx --> Fy) Assumption for C.P.
4. Fb 1,3 MP
5. (Fx & (Hyx --> Fy) 3, Simp.
6. Fb & (Fx & (Hyx --> Fy) Conj. 4, 5
7. Fd MP 6,2
8. Fc & (Fx & (Hyx --> Fy) --> Fd CP 3-7
Having established transitivity, now, go back to Quine’s class treatment and recall the additional
elements, like neckties. Is there an advantage to formulating the ancestral relation in order to avoid
such dross? There are proposals. Let’s take a look at one suggested by Kenneth C. Lucey (Notre
Dame Journal of Formal Logic, Vol. xx. No. 2, April 1979). Lucey will formulate the ancestral in
terms of "generational removal." Now we cannot be, entirely, satisfied inasmuch as the approach
requires a distinction between "direct generational removal" ("DG") and "indirect generational
removal ("NDG"). I will set this aside. In a nutshell the proposal is to build into the "proper ancestral"
the idea of triadic, rather than a dyadic relation: "(En) DG(n, w, z)." Suppose we want to formulate
"w is the great grandfather of z." In this case we would have
DG (3, w, z).
So far so good, but introducing a triadic relation like this raises questions about interpreting the concept
of an ancestor. The problem is that transitivity is affected.. Consider that from
G(1, x, y)
and
G(1, y, z)
we can’t infer
G(1, x,z).
(x and z are not one generatio removed, rather G(3, x, z)
Transitivity fails, but transitivity is part of what is usually part of what is meant by the ancestral relation.
So here is my claim:
If we "squeeze" out the dross introduced by a class approach to the ancestral, the we lose transitivity.
Now this may be a reason to jettison the analysis. I don’t know; I think so, but I’m not going to commit
to this. Instead, I want to point out an application of the following fact:
If we squeeze out the dross introduced by the class interpretation, then we sacrifice transitivity.
At this point I want to move everything in the direction of the logical treatment of causation. I am going
to assume familiarity with the distinction between singularity and regularity theories of causation. Russell
is paradigmatically a regularity theorist’s view; Ducasse is a paradigmatic singularist view.
In treating causation counterfactually D. Lewis introduced the concept of "fragility." Fragility will is
understood, here, as the property which an event has if it could not have taken place differently or at a
different time. (Lewis [1986] p. 196). Elsewhere I defend the view that the more closely two events occur
in time the more fragile they become, but for now I want to make a different point.
Singularity theorists trade in a concept of causation where the events are fragile at the limit. I would prefer
saying that for the singularist events are completely "inelastic." On the view I defend, singular causation is
indicated by the relation which is analogous to the proper ancestral of ‘ancestor’, viz. ‘parent’. Notice that
this is a dyadic relation and intransitive. What I do is reconcile the singularist and regularity approaches by
taking causation in the singularist sense as the proper ancestral of ‘cause’ in the regularity sense. At this point
the "dross" fits in nicely and the usual approach, contra Lucey, is preserved.
Without the "dross" the events held in the causal relation will be inelastic. The class view accommodates, quite
nicely, the elasticity of causation and relates deductively the transitivity of the causal relation to elasticity. One
possible consequence is this: we’ve heard a lot about how probability is part of nature and that causation so
conceived as probabilistic is consistent with "mechanism." (D. Bohm 1957). But I would make the following
point (which coheres well with Bohm’s project): only in the sense that causation is transitive does probability or
chance or randomness enter into a description of causation or lack of it. It is only the transitive sense of ‘cause’ that fits the regularity theory, and this relates to the elasticity of the events in the causal relation . Using the ancestral relation to illustrate the differences between singular and regularity theories helps demonstrate the consistence of these two approaches.
Steve Bayne
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